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Theorem: Every finite dimensional normed space is complete.

$Q$ is a finite-dimensional normed space over $Q$ with the norm $||x||=$ $|x|$. But it is not complete. We have a Cauchy sequence $(1+1/n)^n$ tending to $e\in Q^c$.

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    $\begingroup$ normed spaces are either $\mathbb C$-linear or $\mathbb R$-linear spaces, by definition $\endgroup$ – supinf Aug 21 '17 at 9:41
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Usually a normed vector space is a vector space over the field $K$, where $K= \mathbb R$ or $K= \mathbb C$.

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You are right! In a strange book where "normed space" is defined for fields like $\mathbb Q$ with its usual metric, the theorem should be stated: Every finite-dimensional normed space over a complete field is complete. (I am not aware of such a strange book, however. I have seen "normed space" defined for $p$-adic number fields, which are complete. And maybe an exercise or two with incomplete field, showing why it is not useful in that case.)

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